How many integers $n$ satisfy $(n-2)(n+4)<0$?
Explanation: We consider the signs of the two factors for all possible values of $n$.

If $n>2$, then both $n-2$ and $n+4$ are positive, so the product is positive.

If $n=2$, then $n-2=0$, so the product is 0.

If $-4<n<2$, then $n-2<0$ and $n+4>0$, so the product is negative.

If $n=-4$, then the product is 0.

If $n <-4$, then both factors are negative, and the product is positive.

Therefore, only the integers $-3$, $-2$, $-1$, $0$, and $1$ satisfy the inequality, for a total of $\boxed{5}$.